After quickly recalling what is the Euler characteristic and what is homology and after briefly describing the Jones polynomial, I will move on to describe in simple terms Khovanov's novel "Categorification of the Jones Polynomial", which is to the Jones polynomial as homology is to the Euler characteristic. Namely, it is much more interesting.
With some added and some removed, my talk will be based on my paper On Khovanov's Categorification of the Jones Polynomial and will mostly follow this handout. The only new material will be on the back side of the handout, which will be printed with a list of reasons for why is categorification interestings and with the sketch of the proof that Khovanov's homology lifts to an invariant of knot cobordisms.